Economic Design Of Control Charts Research Articles

Economic design of x ¯ control charts considering process shift distributions

Process shift is an important input parameter in the economic design of control charts. Earlier control chart designs considered constant shifts to occur in the mean of the process for a given assignable cause. This assumption has been criticized by many researchers since it may not be realistic to produce a constant shift whenever an assignable cause occurs. To overcome this difficulty, in the present work, a distribution for the shift parameter has been considered instead of a single value for a given assignable cause. Duncan’s economic design model for chart has been extended to incorporate the distribution for the process shift parameter. It is proposed to minimize total expected loss-cost to obtain the control chart parameters. Further, three types of process shifts namely, positively skewed, uniform and negatively skewed distributions are considered and the situations where it is appropriate to use the suggested methodology are recommended.

Evaluating the performance of variable scheme X-bar control chart: a Taguchi loss approach

The concept of loss function has been combined, with a numerous statistical decision models which require quality cost estimation. Moreover, estimation of the costs of non-conforming product is one of the most important inputs in economic design of control charts. Therefore, on the basis of Markov chain approach, this paper focused on extending control chart with variable schemes control charts by considering Taguchi’s quality loss function. Afterwards, control chart parameters are defined in such a way to minimise the cost of process monitoring via genetic algorithm. Sensitivity analysis on the different sizes of the process shift and the cost of rework or scrap a product and comparing the results unfold insights that will help the quality engineers in designing the most powerful plans for the process.

Edcc: AnRPackage for the Economic Design of the Control Chart

The basic purpose of the economic design of the control charts is to find the optimum control charts parameters to minimize the process cost. In this paper, an R package, <b>edcc</b> (economic design of control charts), which provides a numerical method to find the optimum chart parameters is presented using the unified approach of the economic design. Also, some examples are given to illustrate how to use this package. The types of the control chart available in the <b>edcc</b> package are X?, CUSUM (cumulative sum), and EWMA (exponentially-weighted moving average) control charts.

A parameter-tuned genetic algorithm for statistically constrained economic design of multivariate CUSUM control charts: a Taguchi loss approach

In this research, the main parameters of the multivariate cumulative sum (CUSUM) control chart (the reference value k, the control limit H, the sample size n and the sampling interval h) are determined by minimising the Lorenzen–Vance cost function [Lorenzen, T.J., and Vance, L.C. (1986), ‘The Economic Design of Control Charts: A Unified Approach’, Technometrics, 28, 3–10], in which the external costs of employing the chart are added. In addition, the model is statistically constrained to achieve desired in-control and out-of-control average run lengths. The Taguchi loss approach is used to model the problem and a genetic algorithm, for which its main parameters are tuned using the response surface methodology (RSM), is proposed to solve it. At the end, sensitivity analyses on the main parameters of the cost function are presented and their practical conclusions are drawn. The results show that RSM significantly improves the performance of the proposed algorithm and the external costs of applying the chart, which are due to real-world constraints, do not increase the average total loss very much.

와이블 고장모형 하에서 경고한계를 고려한 관리도의 경제적 설계

Since Duncan(1956) first proposed an economic design of  -control charts, most of the succeeding works on economic design of control charts assumed the exponential failure model like Duncan. Hu(1984), however, assumed a more versatile Weibull failure model to develop an economic design of  -control charts and Banerjee and Rahim(1988) further improved Hu's design by changing the assumption of fixed-length sampling intervals to variable-length ones. In this article we follow the approach of Banerjee and Rahim(1988) but include a pair of warning limits inside the control limits in order to search for a failure without stopping the process when the sample mean falls between warning and control limits. The computational results indicate that the proposed model gives a lower cost than Banerjee and Rahim's model unless the early failure probability of a Weibull distribution is relatively large. The reduction in cost is shown to become larger as the cost of production loss outweighs the cost of searches for a failure.

A hybrid Nelder–Mead simplex and PSO approach on economic and economic-statistical designs of MEWMA control charts

Economic design of a control chart involves determining its basic parameters such that a cost function is minimized. This design when statistical performance measures are also considered is referred to as the economic-statistical design. In this paper, a simplex-based Nelder–Mead algorithm is used in combination with a particle swarm meta-heuristic procedure to solve both the economic and economic-statistical designs of a MEWMA control chart. The application results on extensive simulation experiments show that the particle swarm can lead the Nelder–Mead algorithm to better results. Furthermore, a comparative study is performed on the performances of three different algorithms of the Nelder–Mead, the particle swarm optimization (PSO), and the hybrid PSO and Nelder–Mead (PSO–NM). In this study, five different performance measures are taken into consideration and the results for both the economic and the economic-statistical models are reported at the end.

Economic design of $\bar X$ &amp; S control charts based on Taguchi’s loss function and its optimization

Much research effort has been devoted to economic design of $\bar X$ & S control charts, however, there are some problems in usual methods. On the one hand, it is difficult to estimate the relationship between costs and other model parameters, so the economic design method is often not effective in producing charts that can quickly detect small shifts before substantial losses occur; on the other hand, in many cases, only one type of process shift or only one pair of process shifts are taken into consideration, which may not correctly reflect the actual process conditions. To improve the behavior of economic design of control chart, a cost & loss model with Taguchi’s loss function for the economic design of $\bar X$ & S control charts is embellished, which is regarded as an optimization problem with multiple statistical constraints. The optimization design is also carried out based on a number of combinations of process shifts collected from the field operation of the conventional control charts, thus more hidden information about the shift combinations is mined and employed to the optimization design of control charts. At the same time, an improved particle swarm optimization (IPSO) is developed to solve such an optimization problem in design of $\bar X$ & S control charts, IPSO is first tested for several benchmark problems from the literature and evaluated with standard performance metrics. Experimental results show that the proposed algorithm has significant advantages on obtaining the optimal design parameters of the charts. The proposed method can substantially reduce the total cost (or loss) of the control charts, and it will be a promising tool for economic design of control charts.

Constrained economic design of S control charts for random process shifts

PurposeThe use of control charts to monitor a product quality characteristic requires the selection of their design parameters. To select feasible design parameters, the constraints related to the inspection resources available at the workstation, its configuration and the process operating parameters should be taken into account. The aim of this research is to discuss the design of Shewhart and EWMA control charts monitoring dispersion in the presence of these constraints and for processes characterized by a random shift size.Design/methodology/approachIn this paper the design of the investigated control charts has been intended in a broader sense as the design of a local inspection procedure constrained by a set of resource/environmental process factors characterizing the workstation configuration. An economic objective pursuing the minimization of the total inspection cost has been considered.FindingsConstraining the design of a control chart is an essential issue to find chart design parameters that can be effectively implemented by quality practitioners. The influence of the process‐operating parameters has been modelled and some guidelines have been suggested through the use of contour plots.Originality/valueThe economic design of control charts has been scarcely implemented by quality practitioners due to the difficulty of adapting them to the multifaceted complexity and constraints present within manufacturing environments. This paper tries to contribute to the existing literature by improving the available mathematical models through modelling the actual workstation configuration and resource allocation; furthermore, for random shift processes it investigates the performance of the Shewhart and EWMA control charts monitoring process dispersion.

On the constrained economic design of control charts: a literature review

The economic design is an appealing approach to settle the design parameters of a control chart. Unfortunately, the economic models to design control charts have been scarcely implemented by quality practitioners due to the simplifying assumptions when representing the multifaceted complexity and constraints present within manufacturing and transactional environments. Although there has been an increasing scepticism about the economic models usefulness in practice, some recent studies proposed in literature face the problem of the control charts economic design from a new point of view: the objective is to achieve a well balanced trade-off between the operational and the statistical aspects. Under this perspective, the economic design problem can be intended in a broader sense as the constrained design of a SPC inspection procedure. This paper presents a discussion of some recent trends in the economic design stream of research and outlines the importance of considering the constraints related to SPC resources availability and modelling the occurrence of random shifts.

Economic‐statistical design of variable sampling intervals EWMA charts under non‐normality

PurposeThe purpose of this paper is to investigate the economic‐statistical design of EWMA charts with variable sampling intervals (VSIs) under non‐normality to reduce the process production cycle cost and improve the statistical performance of control charts. The objective is to minimize the cost function by adjusting the control chart parameters which suffice for the statistical restriction.Design/methodology/approachFirst, using the Burr distribution to approximate various non‐normal distributions, the economic‐statistical model of the VSI EWMA charts under non‐normality can be developed. Further, the genetic algorithms will be used to search for the optimal values of parameters of the VSI EWMA charts under non‐normality. Finally, a sensitivity analysis is carried out to investigate the effect of model parameters and statistical restriction on the solution of the economic‐statistical design.FindingsThe result of sensitivity analysis shows that a large lower bound of average time to signal when the process is in control increases the control limit coefficient, no model parameter significantly affects the short sampling intervals, and so on.Originality/valueThe economic‐statistical design method proposed in this paper can improve the statistical performance of economic design of control charts and the general idea can be applied to other VSI control charts.

Economic Design of VSI x-Bar Control Charts for Non Normally Distributed Data under Gamma (λ, 2) Failure Models

This investigation develops the economic design of x-bar control charts with variable sampling interval (VSI) for non normal data under Gamma (λ, 2) shock models. In the past, most of the economic design of control charts follows the Poisson process. However, pragmatically speaking, this process is usually not appropriate. Banerjee and Rahim (1988) presented a cost model which uses variable sampling intervals. This can be contrasted with sampling intervals of fixed length under a process-failure mechanism, which follows a Gamma (λ, 2) model with an increased hazard rate. In a separate study by Al-Oraini and Rahim (2003), an economic statistical design of x-bar control charts with the assumption of the Gamma (λ, 2) failure mechanism was proposed. When the x bar chart is designed to monitor a manufacturing process, three parameters need to be determined via engineers or participators. These parameters are sample size, sampling interval between successive samples, and control limits. However, measurements in the subgroup are assumed to be normally distributed when designing control charts. Said assumption may be untenable. This article employs a numerical example to indicate the solution procedure and to implement sensitivity analysis. Also, the situation of non normal and normal in the Gamma (λ, 2) model was compared. Results in non normal assumption revealed the following: smaller sample size (n) is needed; the initial sampling interval (h 1) can become longer; the sampling interval (h 2) can become shorter; the control limit width (L) can become narrower; and the expected cost per hour (ECT) can likewise be reduced.

Optimal operating policies of X-bar control chart for non-normal variables with generalised Pareto in-control times

In designing the control charts, the choice of control chart parameters like sample size and sampling interval are most important. To determine the optimal design parameters, it is required to consider the model for statistical economic design of control charts with suitable probability distribution for quality characteristic and process in-control times, which are random in nature. In this paper, we consider the economic statistical design of X-bar control chart for non-normal quality characteristic. Here, it is assumed that the sample average of the quality characteristic X-bar has a Johnson distribution and the process in-control times follow generalised Pareto distribution. With suitable cost considerations and modifying the Lorenzen and Vance (1986) cost model, the expected cost per unit time is derived. The optimal sample size and the time interval between samples are obtained by minimising the expected cost per unit time. The sensitivity of the design with respect to the parameters and costs is also studied. This model is much useful for reducing the production cost and to improve product quality in places like, glassware industries, chemicals and films industries, etc.

Economic and economic statistical design of T 2 control chart with two adaptive sample sizes

The Hotelling's T 2 control chart, a direct analogue of the univariate Shewhart X¯ chart, is perhaps the most commonly used tool in industry for simultaneous monitoring of several quality characteristics. Recent studies have shown that using variable sampling size (VSS) schemes results in charts with more statistical power when detecting small to moderate shifts in the process mean vector. In this paper, we build a cost model of a VSS T 2 control chart for the economic and economic statistical design using the general model of Lorenzen and Vance [The economic design of control charts: A unified approach, Technometrics 28 (1986), pp. 3–11]. We optimize this model using a genetic algorithm approach. We also study the effects of the costs and operating parameters on the VSS T 2 parameters, and show, through an example, the advantage of economic design over statistical design for VSS T 2 charts, and measure the economic advantage of VSS sampling versus fixed sample size sampling.

Using economic X control chart design methodology to estimate and optimize machine efficiency in the case of multimachine assignments

AbstractEstimation and optimization of machine efficiency in the case of multi machine assignments is a difficult issue for some industries such as textile industry, due to the machine interference. Although the problem is not new, the solutions proposed are not full satisfactory. For example, the analytical formulas are not suitable for high number of machines; queuing theory approaches are complex and impractical for mill managers. On the other hand, simulation methodology seems to be a perfect tool to estimate and analyze; however, the simulation package programs are expensive and are not suitable for systematic optimization. In this paper, Duncan's economic control chart design methodology is proposed as an alternative way for one of the oldest industrial problems. This study shows that this new approach is a more efficient solution to this stochastic problem. Copyright © 2009 John Wiley &amp; Sons, Ltd.

Economic-Statistical Design of X-bar Control Charts Using Taguchi Loss Functions

Abstract Duncan is the pioneer to provide a control chart design from economic-view point in 1956 and ensure that the economic design of control charts is really lower the cost compared with a conventional Shewhart charts. An economic design does not consider the statistics properties, such as type I or type II error and average time to signal (ATS). In order to improve these issues, an economic-statistical design of control charts was proposed under the statistical consideration. Usually, a product is accepted if the product measurements meet specification requirements and there are no quality losses in the most of economic or economic-statistics design of control charts. Taguchi's Quadratic loss function was employed in this research to construct the quality losses for economic-statistical design of control charts with multiple assignable causes. A numerical example is also used to illustrate its working of the proposed model and to compare the lost cost among the pure economic, economic-statistical control charts design and economic-statistical control charts design using Taguchi's loss function. The example shows that the social loss from the Taguchi's Quadratic loss function is the most important cost compared with the original cost items.

Effect of Preventive Maintenance on the Economic Design of Control Chart

Abstract Economic designs of control charts have been widely investigated and insure that the economic design of control chart actually has a lower cost. This paper presents an integrated model for combining the preventive maintenance and the economic design of control charts. In the proposed model, preventive maintenance can reduce the occurrence rate to an out-of-control state by an amount proportional to the maintenance level on the quality control costs. The maintenance action is conducted on the basis of the sampling results. A numerical example is also used to illustrate the effect of preventive maintenance on the quality control costs and optimum design parameters.

Determination of optimum economic inspection by economic control chart design and by machine efficiency estimation: An application in weaving industry

In this paper, two different approaches are proposed to determine and analyze the optimum economic inspection policy for production quality control. One of these approaches we propose is a reinterpretation of Duncan’s economic and statistical x ¯ control chart design methodology. The other approach is based on a new interpretation of the machine efficiency concept and estimation. It is shown that in this context of application both of these tools used, traditionally used for different purposes, produce similar results. Finally, the study shows that the production efficiency estimation tools can also be used for the production quality control purposes. In both approaches, it is assumed that the deterioration of the considered quality parameters complies with the exponential distribution. The application of these two different approaches is illustrated for the textile weaving industry.

Simultaneous optimization of [Xbar] control chart and age-based preventive maintenance policies under an economic objective

The economic design of control charts and the optimization of preventive maintenance policies have separately received a tremendous amount of attention in the quality and reliability literature over the years in an attempt to reduce the costs associated with operating manufacturing processes. Not until recently has the proposal been made to integrate these two fields and utilize the relationship between quality and equipment performance to improve the productivity of a manufacturing process. In this paper, we extend the initial preliminary investigation of this idea of using an chart in conjunction with an age-replacement preventive maintenance policy. We formulate a partially observable, discrete-time Markov decision process in order to obtain the near-optimal combined preventive maintenance/statistical process control policy that minimizes the costs associated with maintenance, sampling, and poor quality. We develop transition probabilities for the various states of the infinite horizon problem and a solution algorithm for finding the best policy in polynomial time by finding a control limit on the sampling policy. We also perform sensitivity analysis on the decision variables for each of the various input parameters. It is shown that in every case a combined PM and SPC policy is the most cost efficient.

Design of EWMA and CUSUM control charts subject to random shift sizes and quality impacts

Statistical process control charts are important tools for detecting process shifts. To ensure accurate, responsive fault detection, control chart design is critical. In the literature, control charts are typically designed by minimizing the control chart's responding time, i.e., average run length (ARL), to an anticipated shift size under a tolerable false alarm rate. However, process shifts, originating from various variation sources, often come with different sizes and result in different degrees of quality impacts. In this paper, we propose a new performance measure for EWMA and CUSUM control chart design to take into consideration the variable shift sizes and corresponding quality impacts. Unlike economic designs of control charts that suffer from a complex cost structure and intensive numerical computation, the proposed design methodology does not involve any cost estimation and the design procedure is as simple as looking up tables. Given the Gaussian random shifts and quadratic quality loss function, we show that the proposed design has a significant reduction in the quality impact as compared to conventional ARL-based designs. Guidelines and useful worksheets for practical implementation of the proposed designs are then suggested to practitioners with different knowledge levels of the process excursions.

Using economically designed Shewhart and adaptive ―X charts for monitoring the quality of tiles

AbstractThis paper presents the economic design of ―X control charts for monitoring a critical stage of the main production process at a tile manufacturer in Greece. Two types of ―X charts were developed: a Shewhart‐type chart with fixed parameters and adaptive charts with variable sampling intervals and/or sample size. Our prime motivation was to improve the statistical control scheme employed for monitoring an important quality characteristic of the process with the objective of minimizing the relevant costs. At the same time we tested and confirmed the applicability of the theoretical models supporting the economic design of control charts with fixed and variable parameters in a practical situation. We also evaluated the economic benefits of moving from the broadly used static charts to the application of the more flexible and effective adaptive control charts. The main result of our study is that, by redesigning the currently employed Shewhart chart using economic criteria, the quality‐related cost is expected to decrease by approximately 50% without increasing the implementation complexity. Monitoring the process by means of an adaptive ―X chart with variable sampling intervals will increase the expected cost savings by about 10% compared with the economically designed Shewhart chart at the expense of some implementation difficulty. Copyright © 2006 John Wiley &amp; Sons, Ltd.